Show that if $\displaystyle x_n $ and $\displaystyle y_n $ are Cauchy sequences then both $\displaystyle x_n+y_n $ and $\displaystyle x_n y_n $ are Cauchy sequences
The first is simple noting $\displaystyle \left| {(x_m + y_m ) - (x_n + y_n )} \right| \leqslant \left| {x_m - x_n } \right| + \left| {y_m - y_n } \right|$.
For the second we know that Cauchy sequences are bounded.
$\displaystyle \left| {(x_m y_m ) - (x_n y_n )} \right| \leqslant \left| {(x_m y_m ) - (x_n y_m )} \right| + \left| {(x_n y_m ) - (x_n y_n )} \right|$$\displaystyle \leqslant \left| {y_m } \right|\left| {x_m - x_n } \right| + \left| {x_n } \right|\left| {y_m - y_n } \right|$